This is quite a complex problem. I wrote out a really nice solution but I can't work out how to put it on the website as the app is very poorly made. Still, I'll just have to type it all in...
Okay so you need to use a technique called logarithmic differentiation. It seems quite unnatural to start with but the result is very impressive.
Let y = (x+8)^(3x)
Take the natural log of both sides:
ln(y) = ln((x+8)^(3x))
By laws of logarithms, this can be rearranged:
ln(y) = 3xln(x+8)
Next, differentiate both sides. By implicit differentiation:
d/dx(ln(y)) = 1/y dy/dx
The right hand side is harder to differentiate. Using the substitution u = 3x and v = ln(x+8):
d/dx(3xln(x+8)) = d/dx(uv)
du/dx = 3
Finding dv/dx is harder, and involves the chain rule. Let a = x+ 8:
v = ln(a)
da/dx = 1
dv/da = 1/a
By chain rule:
dv/dx = dv/da * da/dx = 1/a = 1/(x+8)
Finally, use the product rule:
d/dx(uv) = u * dv/dx + v * du/dx = 3x/(x+8) + 3ln(x+8)
This overall produces the equation:
1/y * dy/dx = 3x/(x+8) + 3ln(x+8)
We want to solve for dy/dx, achievable by multiplying both sides by y:
dy/dx = y(3x/(x+8) + 3ln(x+8))
Since we know y = (x+8)^(3x):
dy/dx = ((x+8)^(3x))(3x/(x+8) + 3ln(x+8))
Neatening this up a bit, we factorise out 3/(x+8):
dy/dx = (3(x+8)^(3x-1))(x + (x+8)ln(x+8))
Well wasn't that a marathon? It's a nightmare typing that in, I hope you can follow all the steps.
I hope this helped you :)
An equation is formed of two equal expressions. The value of the constant of proportionality is 0.41.
<h3>What is an equation?</h3>
An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Given there exists a proportional relationship between the number of juice bottles bought, j, and the total cost in dollars and cents. Therefore,
c ∝ j
c = kj
As the relation is represented by the equation c=0.41j. Therefore, the value of the constant of proportionality is 0.41.
Learn more about Equation:
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Answer:
b.a rotation 270 clockwise
Step-by-step explanation:
To answer your question, the answer would be 8
264/22 = 12 that would be the same and Omarion's painting but for it to be a better deal, it would be 13 hours.
13 hours <span>Omarion's painting : $273 (Better deal)
13 hours </span><span> Colorful paints : $286 </span>
